Abstract
Superhydrophobic surfaces can significantly reduce hydrodynamic skin drag by accommodating large slip velocity near the surface due to entrapment of air bubbles within their micro-scale roughness elements. While there are many Stokes flow solutions for flows near superhydrophobic surfaces that describe the relation between effective slip length and surface geometry, such relations are not fully known in the turbulent flow limit. In this work, we present a phenomenological model for the kinematics of flow near a superhydrophobic surface with periodic post-patterns at high Reynolds numbers. The model predicts an inverse square root scaling with solid fraction, and a cube root scaling of the slip length with pattern size, which is different from the reported scaling in the Stokes flow limit. A mixed model is then proposed that recovers both Stokes flow solution and the presented scaling, respectively, in the small and large texture size limits. This model is validated using direct numerical simulations of turbulent flows over superhydrophobic posts over a wide range of texture sizes from L+ ≈ 6 to 310 and solid fractions from ϕs = 1/9 to 1/64. Our report also embarks on the extension of friction laws of turbulent wall-bounded flows to superhydrophobic surfaces. To this end, we present a review of a simplified model for the mean velocity profile, which we call the shifted-turbulent boundary layer model, and address two previous shortcomings regarding the closure and accuracy of this model. Furthermore, we address the process of homogenization of the texture effect to an effective slip length by investigating correlations between slip velocity and shear over pattern-averaged data for streamwise and spanwise directions. For L+ of up to O(10), shear stress and slip velocity are perfectly correlated and well described by a homogenized slip length consistent with Stokes flow solutions. In contrast, in the limit of large L+, the pattern-averaged shear stress and slip velocity become uncorrelated and thus the homogenized boundary condition is unable to capture the bulk behavior of the patterned surface.
Highlights
Where us is the averaged velocity on the boundary, called the slip velocity, and ∂u/∂n is the wall-normal derivative of the mean velocity profile and represents the mean shear when multiplied by the fluid viscosity. b is the effective slip length, which depends solely on the texture geometry in the Stokes flow limit
The near wall region of turbulent flows is controlled by the shear length scale, δν, which is much smaller than the macroscopic geometry inversely proportional to the Reynolds number
We presented a comprehensive investigation of the kinematic properties of turbulent flows over superhydrophobic surfaces with micro-posts
Summary
Where us is the averaged velocity on the boundary, called the slip velocity, and ∂u/∂n is the wall-normal derivative of the mean velocity profile and represents the mean shear when multiplied by the fluid viscosity. b is the effective slip length, which depends solely on the texture geometry in the Stokes flow limit. Reducing the drag by only 30% in a laminar channel flow would require a SHS texture size of about 15% of channel height with φs ≈ 10% With such design, the percentage of drag reduction remains independent of the flow speed as long as the flow remains laminar. The near wall region of turbulent flows is controlled by the shear length scale, δν, which is much smaller than the macroscopic geometry inversely proportional to the Reynolds number. For the aforementioned example of a channel flow, the same 30% drag reduction would require
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